Multiplying a conjugacy class by its inverse in a finite group

Abstract

Suppose that $G$ is a finite group and $K$ a non-trivial conjugacy class of $G$ such that $K{K}^{-1}=1\cup D\cup D^{-1}$ with $D$ a conjugacy class of $G$. We prove that $G$ is not a non-abelian simple group. We also give arithmetical conditions on the class sizes determining the structure of $⟨K⟩$ and $⟨D⟩$. Furthermore, if $D=K$ is a non-real class, then $⟨K⟩$ is $p$-elementary abelian for some odd prime $p$.